Natural Numbers

All the positive integers from 1, 2, 3,……, ∞.

Whole Numbers

All the natural numbers including zero are called Whole Numbers.

Integers

All negative and positive numbers including zero are called Integers.

Rational Numbers

 A number is called Rational if it can be expressed in the form p/q where p and q are integers

 (q> 0). It includes all natural, whole number and integers.Example: 1/2, 4/3, 5/7,1 etc. 

1.2  PROPERTIES OF RATIONAL NUMBERS:

  1. Closure Property

This shows that the operation of any two same types of numbers is also the same type or not.

a). Whole Numbers

If p and q are two whole numbers then

Operation Addition Subtraction Multiplication Division

 

Whole number

p + q will also be the whole number.

p – q will not always be a whole number.

pq will also be the whole number.

p ÷ q will not always be a whole number.

Example

6 + 0 = 6

8 – 10 = – 2

3 × 5 = 15

3 ÷ 5 = 3/5

Closed or Not

Closed

Not closed

Closed

Not closed

b). Integers

If p and q are two integers then

Operation

Addition

Subtraction

Multiplication

Division

Integers

p+q will also be an integer.

p-q will also be an integer.

pq will also be an integer.

p ÷ q will not always be an integer.

Example - 3 + 2 = – 1 5 – 7 = – 2  - 5 × 8 = – 40 - 5 ÷ 7  = – 5/7
Closed or not Closed Closed Closed Not  closed

 

c). Rational Numbers

If p and q are two rational numbers then

Operation

Addition

Subtraction

Multiplication

Division

Rational Numbers p + q will also be a rational number. p – q will also be a rational number. pq will also be a rational number. p ÷ q will not always be a rational number
Example p ÷ 0

= not defined

Closed or not

Closed

Closed

Closed

Not  closed

 

  1. Commutative Property

This shows that the position of numbers does not matter i.e. if you swap the positions of the numbers then also the result will be the same.

a). Whole Numbers

If p and q are two whole numbers then 

Operation Addition Subtraction Multiplication Division
Whole number p + q = q + p p – q ≠ q – p  p × q = q × p p ÷ q ≠ q ÷ p
Example 3 + 2 = 2 + 3 8 –10 ≠ 10 – 8 – 2 ≠ 2 3 × 5 = 5 × 3 3 ÷ 5 ≠ 5 ÷ 3
Commutative yes No yes No 

 

b). Integers

If p and q are two integers then

 

Operation Addition Subtraction Multiplication Division
Whole number p + q = q + p p – q ≠ q – p  p × q = q × p p ÷ q ≠ q ÷ p
Example True 5 – 7 = – 7 – (5) - 5 × 8 = 8 × (–5) - 5 ÷ 7 ≠ 7 ÷ (-5)
Commutative yes No yes No 

 

c). Rational Numbers

If p and q are two rational numbers then

Operation Addition Subtraction Multiplication Division
Example  

=   + (

 

 

 (   )  - 

 

 

=

 

 

 

 

 

Example True 5 – 7 = – 7 – (5) - 5 × 8 = 8 × (–5) - 5 ÷ 7 ≠ 7 ÷ (-5)
Commutative yes No yes No 

 

  1. Associative Property

This shows that the grouping of numbers does not matter i.e. we can use operations on any two numbers first and the result will be the same.

a). Whole Numbers

If p, q and r are three whole numbers then

 

Operation Addition Subtraction Multiplication Division
Whole number p + (q + r) = (p + q) + r p – (q – r) = (p – q) – r p × (q × r) = (p × q) × r p ÷ (q ÷ r)  ≠ (p ÷ q) ÷ r
Example 3 + (2 + 5) = (3 + 2) + 5 8 – (10 – 2) ≠ (8 -10) – 2 3 × (5 × 2) = (3 × 5) × 2 10 ÷ (5 ÷ 1) ≠ (10 ÷ 5) ÷ 1
Associative yes No yes No 

 

  1. Integers

If p, q and r are three integers then

Operation Integers Example Associative
Addition p + (q + r) = (p + q) + r (– 6) + [(– 4)+(–5)] = [(– 6) +(– 4)] + (–5) Yes
Subtraction p – (q – r) = (p – q) – r 5 – (7 – 3) ≠ (5 – 7) – 3 No
Multiplication p × (q × r) = (p × q) × r (– 4) × [(– 8) ×(–5)] = [(– 4) × (– 8)] × (–5) Yes
Division
p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r

 

[(–10) ÷ 2] ÷ (–5) ≠ (–10) ÷ [2 ÷ (– 5)] No

 

  1. Rational Numbers

If p, q and r are three rational numbers then

Operation Integers Example Associative
Addition p + (q + r) = (p + q) + r   =

=  =

 

 

Yes

Subtraction p – (q – r) = (p – q) – r No
Multiplication p × (q × r) = (p × q) × r  =  ×

Hence LHS = RHS

Yes
Division p ÷ (q ÷ r)  ≠ (p ÷ q) ÷ r  

 

 

 

 

No

The Role of Zero in Numbers (Additive Identity)

Zero is the additive identity for whole numbers, integers and rational numbers.

  Identity   Example
Whole number a + 0 = 0 + a = a Addition of zero to whole number  2 + 0 = 0 + 2 = 2
Integer b + 0 = 0 + b = b Addition of zero to an integer  False
Rational number c + 0 = 0 + c = c Addition of zero to a rational number  2/5 + 0 = 0 + 2/5 = 2/5

 

The Role of one in Numbers (Multiplicative Identity)  

One is the multiplicative identity for whole numbers, integers and rational numbers.

  Identity   Example
Whole number a ×1 = a Multiplication of one to the whole number 5 × 1 = 5
Integer b × 1= b Multiplication of one to an integer - 5 × 1 = – 5
Rational Number c × 1= c Multiplication of one to a rational number  

 

Negative of a Number (Additive Inverse)

 

  Identity   Example
Whole number a +(- a) = 0 Where a is a  whole number  5 + (-5) = 0
Integer b +(- b) = 0 Where b is an integer  True
Rational number c + (-c) = 0 Where c is a rational number  

 

 

Reciprocal (Multiplicative Inverse)

The multiplicative inverse of any rational number

Example :The reciprocal of  is.

Distributivity of Multiplication over Addition and Subtraction for Rational Numbers

This shows that for all rational numbers p, q and r

  1. p(q + r) = pq + pr2. p(q – r) = pq – pr

Example:Check the distributive property of the three rational numbers 4/7,-( 2)/3 and 1/2.

Solution:Let’s find the value of